Geometric Method: A Novel, Fast and Accurate Solution for the Inverse Problem in Risley Prisms

Abstract

Today, mechanical tracking systems are becoming increasingly compact, enabling a new range of civil and military applications. These include aerial laser scanning, for which Risley prisms are used. In Risley systems, the so-called inverse problem, which focuses on obtaining the angles of the prisms for a given target coordinate, has not yet been solved mathematically. As a consequence, approximate approaches have been used, but the solutions obtained have significant errors and a lack of precision. To improve accuracy, iterative methods, which are computationally intensive, have also been implemented. In this paper, an analytical process which we call the geometric method is presented, and we verified that this strategy highly improves accuracy and computational speed. Using this method in an iterative process gives accuracies of up to 1 pm in only three iterations. This high accuracy would allow the geometric method to be applied in fields such as lithography, stereolithography, or 3D printing.